for any measurable subset \(\mathcal {A} \subset \mathbb {O}(n)\)  and any fixed \(U\in \mathbb {O}(n)\) . Here, \(U\circ \mathcal {A} \overset{\text{def}}{=}\left\lbrace UV: V\in \mathcal {A}\right\rbrace \)  and \(\mathcal {A} \circ U\)  is defined similarly. The classical Haar's theorem [1]}, [2]} shows that there is one, and only one, translation-invariant probability measure in the sense of (REF ) on \(\mathbb {O}(n)\) . In fact, the theorem holds in much greater generality. For example, it remains true for any compact Lie group, which includes \(\mathbb {O}(n)\)  [and \(\mathbb {U}(n)\) ] as its special case.
